多変量正規分布のモーメントとキュムラント

\begin{align}\mathrm{E}[X_h]&=\left.\cfrac{1}{i}\cfrac{\partial(\boldsymbol{t})}{\partial t_h}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=\left.\cfrac{1}{i}\left\{e^{i\boldsymbol{t}^T\boldsymbol{\mu}-\frac{1}{2}\boldsymbol{t}^T\boldsymbol{\Sigma t}}\cfrac{\partial}{\partial t_h}\left(i\boldsymbol{t}^T\boldsymbol{\mu}-\cfrac{1}{2}\boldsymbol{t}^T\boldsymbol{\Sigma t}\right)\right\}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=\left. \cfrac{1}{i}\left\{\phi(\boldsymbol{t})\cfrac{\partial }{\partial t_h}\left(\sum_{j=1}^p it_j\mu_j-\cfrac{1}{2}\sum_{j,k=1}^pt_jt_k\sigma_{jk}\right)\right\}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=\left.\cfrac{1}{i}\left\{\phi(\boldsymbol{t})\left(i\mu_h-\cfrac{1}{2}2\sum_{j=1}^pt_j\sigma_{hj}\right)\right\}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=\cfrac{1}{i}\left\{1\cdot \left(i\mu_h-\sum_{j=1}^p0\sigma_{hj}\right)\right\}\\\label{eq1}&=\mu_h.\tag{1}\end{align}

\begin{align}\mathrm{E}[X_hX_j]&=\left.\cfrac{1}{i^2}\cfrac{\partial^2}{\partial t_h\partial t_j}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-\left.\cfrac{\partial}{\partial t_j}\left\{\left(i\mu_h-\sum_{j=1}^pt_j\sigma_{hj}\right)\phi(\boldsymbol{t})\right\}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-\left.\left\{\cfrac{\partial}{\partial t_j}\left(i\mu_h-\sum_{j=1}^pt_{j}\sigma_{hj}\right)+\left(i\mu_h-\sum_{j=1}^pt_j\sigma_{hj}\right)\cfrac{\partial}{\partial t_j}\phi(\boldsymbol{t})\right\}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-\left.\left\{-\sigma_{hj}\phi(\boldsymbol{t})+\left(i\mu_h-\sum_{j=1}^pt_j\sigma_{hj}\right)\left(i_mu_j-\sum_{k=1}^pt_k\sigma_{jk}\right)\phi(\boldsymbol{t})\right\}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-\left.\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{j=1}^pt_j\sigma_{hj}\right)\left(i_mu_j-\sum_{k=1}^pt_k\sigma_{jk}\right)\right\}\phi(\boldsymbol{t})\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-\bigl\{\sigma_{hj}+(0+i\mu_h)(0+i\mu_j)\bigr\}\cdot1\\\label{eq2}&=\sigma_{hj}+\mu_h\mu_j.\tag{2}\end{align}

\begin{align}\mathrm{var}[X_i]&=\mathrm{E}\bigl[(X_i-\mu_i)^2\bigr]\notag\\&=\mathrm{E}[X_i^2]-2\mu_i\mathrm{E}[X_i]+\mu_i^2\\&=\sigma_{ii}+\mu_i^2-2\mu_i^2+\mu_i^2\sigma_{ii}\\\label{eq3}&=\sigma_{ii}\tag{3}\end{align}

\begin{align}\mathrm{Cov}[X_i, X_j] &= \mathrm{E}\bigl[(X_i-\mu_i)(X_j-\mu_j)\bigr]\\&=\mathrm{E}[X_iX_j]-\mu_j\mathrm{E}[X_i]-\mu_i\mathrm{E}[X_j]+\mu_i\mu_j\\\label{eq4}&=\sigma_{ij}+\mu_i\mu_j-\mu_i\mu_j-\mu_i\mu_j+\mu_i\mu_j\\&=\sigma_{ij}.\tag{4}\end{align}

\begin{align}\mathrm{E}[X_hX_jX_k] &= \left.\cfrac{1}{i^4}\cfrac{\partial^3 \phi(\boldsymbol{t})}{\partial t_h\partial t_j\partial t_k}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-i\left.\cfrac{\partial}{\partial t_k}\left[\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{l=1}^pt_l\sigma_{hl}\right)\left(i\mu_j-\sum_{l=1}^pt_l\sigma_{jl}\right)\right\}\phi(\boldsymbol{t})\right]\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-i\Biggl[\cfrac{\partial}{\partial t_k}\left\{\sigma_{hj}+\left(i\mu_h-\sum_{l=1}^pt_l\sigma_{hl}\right)\left(i\mu_j-\sum_{l=1}^pt_l\sigma_{jl}\right)\right\}\phi(\boldsymbol{t}) \\&\ \ \ \ \left.\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{l=1}^pt_l\sigma_{hl}\right)\left(i\mu_j-\sum_{l=1}^pt_l\sigma_{jl}\right)\right\}\cfrac{\partial}{\partial t_k}\phi(\boldsymbol{t})\Biggr]\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-i\Biggl[\cfrac{\partial}{\partial t_k}\left\{-\sigma_{hj}+\sum_{l,m=1}^pt_lt_m\sigma_{hl}\sigma_{jm}-i\mu_j\sum_{l=1}^pt_l\sigma_{hl}-i\mu_h\sum_{l=1}^pt_l\sigma_{jl}-\mu_h\mu_j\right\}\phi(\boldsymbol{t})\\&\ \ \ \ +\left.\left\{\left(i\mu_h-\sum_{l=1}^pt_l\sigma_{hl}\right)\left(i\mu_j-\sum_{l=1}^pt_l\sigma_{jl}\right)\left(i\mu_k-\sum_{l=1}^pt_l\sigma_{kl}\right)\right\}\phi(\boldsymbol{t})\Biggr]\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-i\Biggl[\cfrac{\partial}{\partial t_k}\Biggl\{-\sigma_{hj}+\sum_{l=1}^pt_lt_k\sigma_{hl}\sigma_{jk}+\sum_{l=1}^pt_ht_k\sigma_{hk}\sigma_{jl}\\&\ \ \ \ +\sum_{l,m\neq k}^pt_lt_m\sigma_{hl}\sigma_{jm}-i\mu_j\sum_{l=1}^pt_l\sigma_{jl}-i\mu_h\sum_{l=1}^pt_l\sigma_{jl}-\mu_h\mu_j\Biggr\}\phi(\boldsymbol{t})\\&\ \ \ \ +\left.\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{l=1}^pt_l\sigma_{hl}\right)\left(i\mu_j-\sum_{l=1}^pt_l\sigma_{jl}\right)\right\}\left(i\mu_k-\sum_{l=1}^pt_l\sigma_{kl}\right)\phi(\boldsymbol{t})\Biggr]\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-i\Biggl\{-0+\sum_{l,m=1}^pt_{l}\sigma_{hk}\sigma_{jl}+\sum_{l,m=1}^pt_{l}\sigma_{hl}\sigma_{jk}+0-i\mu_j\sigma_{hk}-i\mu_h\sigma_{jk}-0\Biggr\}\phi(\boldsymbol{t})\\&\ \ \ \ +\left.\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{l=1}^pt_l\sigma_{hl}\right)\left(i\mu_j-\sum_{l=1}^pt_l\sigma_{jl}\right)\right\}\left(i\mu_k-\sum_{l=1}^pt_l\sigma_{kl}\right)\phi(\boldsymbol{t})\Biggr]\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=-i\left\{(i\mu_j\sigma_{hk}-i\mu_h\sigma_{jk})\cdot1 +i\mu_k(-\mu_h\mu_j-\sigma_{hj})\cdot1\right\}\\&=\mu_h\mu_j\mu_k+\mu_h\sigma_{kj}+\mu_j\sigma_{hk}+\mu_k\sigma_{hj}. \end{align}

\begin{align}&\mathrm{E}\bigl[(X_h-\mu_h)(X_j-\mu_j)(X_k-\mu_k)\bigr]\\&=\mathrm{E}[X_hX_jX_k-X_hX_j\mu_k-X_hX_k\mu_j-X_jX_k\mu_h+X_h\mu_j\mu_k+X_j\mu_h\mu_k+X_h\mu_j\mu_k-\mu_h\mu_j\mu_k]\\&=\mu_h\mu_j\mu_k+\mu_h\sigma_{kj}+\mu_j\sigma_{hk}+\mu_k\sigma_{hj}\\&\ \ \ \ -(\sigma_{hj}+\mu_h\mu_j)\mu_k-(\sigma_{hk}+\mu_h\mu_k)\mu_j-(\sigma_{jk}+\mu_j\mu_k)\mu_h-\mu_h\mu_j\mu_k\\\label{eq5}&=0.\tag{5}\end{align}

\begin{align}& \mathrm{E}[X_hX_jX_kX_l] \\ &=\left.\cfrac{1}{i^4}\cfrac{\partial^4}{\partial t_h\partial t_j\partial t_k\partial t_l}\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=\Biggl[\cfrac{\partial}{\partial t_l}\Biggl\{\sum_{m,n=1}^pt_{m}\sigma_{hm}\sigma_{jn}+\sum_{m,n=1}^pt_{n}\sigma_{hm}\sigma_{jn}-i\mu_j\sigma_{hk}-i\mu_h\sigma_{jk}\\&\ \ \ \ +\left.\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{m=1}^pt_m\sigma_{hm}\right)\left(i\mu_j-\sum_{m=1}^pt_m\sigma_{jm}\right)\right\}\left(i\mu_k-\sum_{m=1}^pt_m\sigma_{km}\right)\phi(\boldsymbol{t})\Biggr\}\Biggl]\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=\Biggl[\cfrac{\partial}{\partial t_l}\Biggl\{\sum_{m,n=1}^pt_{m}\sigma_{hm}\sigma_{jn}+\sum_{m,n=1}^pt_{n}\sigma_{hm}\sigma_{jn}-i\mu_j\sigma_{hk}-i\mu_h\sigma_{jk}\\&\ \ \ \ +\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{m=1}^pt_m\sigma_{hm}\right)\left(i\mu_j-\sum_{m=1}^pt_m\sigma_{jm}\right)\right\}\left(i\mu_k-\sum_{m=1}^pt_m\sigma_{km}\right)\Biggr\}\phi(\boldsymbol{t})\\&\ \ \ \  +\Biggl\{\sum_{m,n=1}^pt_{m}\sigma_{hm}\sigma_{jn}+\sum_,{m,n=1}^pt_{n}\sigma_{hm}\sigma_{jn}-i\mu_j\sigma_{hk}-i\mu_h\sigma_{jk}\\&\ \ \ \ +\left.\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{m=1}^pt_m\sigma_{hm}\right)\left(i\mu_j-\sum_{m=1}^pt_m\sigma_{jm}\right)\right\}\left(i\mu_k-\sum_{m=1}^pt_m\sigma_{km}\right)\Biggr\}\cfrac{\partial}{\partial t_l}\phi(\boldsymbol{t})\Biggr]\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=\Biggl[\cfrac{\partial}{\partial t_l}\Biggl\{\sum_{m,n=1}^pt_{m}\sigma_{hm}\sigma_{jn}+\sum_{m,n=1}^pt_{n}\sigma_{hm}\sigma_{jn}-i\mu_j\sigma_{hk}-i\mu_h\sigma_{jk}\\&\ \ \ \ +\Biggl\{\sum_{m,n,o=1}^p -t_m t_n t_o\sigma_{hm}\sigma_{jn}\sigma_{ko}+i\mu_j\sum_{m,n=1}^pt_mt_n\sigma_{hm}\sigma_{kn}+i\mu_h\sum_{m,n=1}^pt_mt_n\sigma_{jm}\sigma_{kn}+\mu_h\mu_j\sum_{m=1}^pt_m\sigma_{km}\\&\ \ \ \ +\sigma_{hj}\sum_{m=1}^pt_m\sigma_{km}\Biggr\}+i\mu_k\Biggl\{\sum_{m,n=1}^p t_mt_n\sigma_{hm}\sigma_{jn}-i\mu_j\sum_{m=1}^pt_m\sigma_{hm}-i\mu_h\sum_{m=1}^pt_m\sigma_{jm}-\mu_h\mu_j-\sigma_{hj}\Biggr\}\Bigg\}\phi(\boldsymbol{t})\\&\ \ \ \ +\Biggl\{\sum_{m,n=1}^pt_{m}\sigma_{hm}\sigma_{jn}+\sum_{m,n=1}^pt_{n}\sigma_{hm}\sigma_{jn}-i\mu_j\sigma_{hk}-i\mu_h\sigma_{jk}\\&\ \ \ \ +\left.\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{m=1}^pt_m\sigma_{hm}\right)\left(i\mu_j-\sum_{m=1}^pt_m\sigma_{jm}\right)\right\}\left(i\mu_k-\sum_{m=1}^pt_m\sigma_{km}\right)\Biggr\}\cfrac{\partial}{\partial t_l}\phi(\boldsymbol{t})\Biggr]\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=\Biggl\{(\sigma_{hl}\sigma_{jk}+\sigma_{hk}\sigma_{jl})+\left(-\sum_{m,n\neq l}^p t_m t_n\sigma_{hm}\sigma_{jn}\sigma_{kl}-\sum_{m,n\neq l}^p t_mt_n\sigma_{hm}\sigma_{jl}\sigma_{kn}-\sum_{m,n\neq l}^p t_mt_n\sigma_{hl}\sigma_{jm}\sigma_{kn}\right)\\&\ \ \ \ -2\left(\sum_{m\neq l}^pt_mt_l\sigma_{hm}\sigma_{jl}\sigma_{kl}+\sum_{m\neq l}^pt_mt_l\sigma_{hl}\sigma_{jm}\sigma_{kl}\sum_{m\neq l}^pt_mt_l\sigma_{hl}\sigma_{jl}\sigma_{km}\right)-3t_l^2\sigma_{hl}\sigma_{jl}\sigma_{kl}\\&\ \ \ \ +i\mu_j\Bigl(\sum_{m=1}^pt_m \sigma_{hm}\sigma_{kl} +\sum_{m=1}^p t_m\sigma_{hl}\sigma_{jm}\Bigr)\\&\ \ \ \ +i\mu_h\Bigl(\sum_{m=1}^pt_m\sigma_{jm}\sigma_{kl}+\sum_{m=1}^pt_m\sigma_{jl}\sigma_{km}\Bigr)+\mu_h\mu_j\sigma_{kl}+\sigma_{hj}\sigma_{kl}\\&\ \ \ \ +i\mu_k\Bigl(\sum_{m=1}^pt_m\sigma_{hm}\sigma_{jl}+\sum_{m=1}^pt_m\sigma_{hl}\sigma_{jm}-i\mu_j\sigma_{hl}-i\mu_h\sigma_{jl}\Bigr)\Biggr\}\phi(\boldsymbol{t})\\&\ \ \ \ +\Biggl[\Biggl\{\sum_{m,n=1}^pt_{m}\sigma_{hm}\sigma_{jn}+\sum_{m,n=1}^pt_{n}\sigma_{hm}\sigma_{jn}-i\mu_j\sigma_{hk}-i\mu_h\sigma_{jk}\\&\ \ \ \ +\left\{-\sigma_{hj}+\left(i\mu_h-\sum_{m=1}^pt_m\sigma_{hm}\right)\left(i\mu_j-\sum_{m=1}^pt_m\sigma_{jm}\right)\right\}\left(i\mu_k-\sum_{m=1}^pt_m\sigma_{km}\right)\Biggr\}\cfrac{\partial}{\partial t_l}\phi(\boldsymbol{t})\Biggr]\\&\ \ \ \ \times\left. \left(i\mu_l-\sum_{m=1}^pt_m\sigma_{lm}\right)\phi(\boldsymbol{t})\Biggr]\right|_{\boldsymbol{t}=\boldsymbol{0}}\\&=\bigl\{(\sigma_{hl}\sigma_{jk}+\sigma_{hk}\sigma_{jl})+\mu_h\mu_j\sigma_{kl} + \sigma_{hj}\sigma_{kl}+i\mu_k(-i\mu_j\sigma_{hl}-i\mu_h\sigma_{jl})\bigr\}\cdot1\\&\ \ \ \ +\bigl[-i\mu_j\sigma_{hk}-i\mu_h\sigma_{jk}+i\mu_k(-\mu_h\mu_j-\sigma_{hj})\bigr]i\mu_l\cdot1\\&=\sigma_{hj}\sigma_{kl}+\sigma_{hk}\sigma_{jl}+\sigma_{hl}\sigma_{jk}+\mu_h\mu_j\sigma_{kl}+\mu_j\mu_k\sigma_{hl}+\mu_h\mu_k\sigma_{jl}\\&\ \ \ \ +\mu_h\mu_j\mu_k\mu_l+\mu_k\mu_l\sigma_{hj}+\mu_h\mu_l\sigma_{jk}+\mu_j\mu_l\sigma_{hk}.\end{align}

\begin{align}&\mathrm{E}\bigl[(X_h-\mu_h)(X_j-\mu_j)(X_k-\mu_k)(X_l-\mu_l)\bigr]\\&=\mathrm{E}[X_hX_jX_kX_l-X_hX_jX_k\mu_l-X_hX_jX_l\mu_k+X_hX_j\mu_k\mu_l-X_hX_kX_l\mu_j+X_hX_k\mu_j\mu_l+X_hX_l\mu_j\mu_k\\&\ \ \ \ -X_h\mu_j\mu_k\mu_l-X_jX_kX_l\mu_h+X_jX_k\mu_h\mu_l+X_jX_l\mu_h\mu_k-X_j\mu_h\mu_k\mu_l+X_kX_l\mu_h\mu_j-X_k\mu_h\mu_j\mu_l\\&\ \ \ \ -X_l\mu_h\mu_j\mu_k+\mu_h\mu_j\mu_k\mu_l]\\&=\sigma_{hj}\sigma_{kl}+\sigma_{hk}\sigma_{jl}+\sigma_{hl}\sigma_{jk}+\mu_h\mu_j\mu_k\mu_l-4\mu_h\mu_j\mu_k\mu_l+6\mu_h\mu_j\mu_k\mu_l-3\mu_h\mu_j\mu_k\mu_l\\&\ \ \ \ +2(\mu_h\mu_j\sigma_{kl}+\mu_j\mu_k\sigma_{hl}+\mu_h\mu_k\sigma_{jl}+\mu_k\mu_l\sigma_{hj}+\mu_h\mu_l\sigma_{jk}+\mu_j\mu_l\sigma_{hk})\\&\ \ \ \ -2(\mu_h\mu_j\sigma_{kl}+\mu_j\mu_k\sigma_{hl}+\mu_h\mu_k\sigma_{jl}+\mu_k\mu_l\sigma_{hj}+\mu_h\mu_l\sigma_{jk}+\mu_j\mu_l\sigma_{hk})\\\label{eq6}&=\sigma_{hj}\sigma_{kl}+\sigma_{hk}\sigma_{jl}+\sigma_{hl}\sigma_{jk}.\tag{6}\end{align}